Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C2 |
| direct product | C17×#7542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1415 |
| commutator | C32 | C2×C4×C17 |
| Frattini, cosocle | C2×C32 | C22×C17 |
| prime radical | C22×C17 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×C8×C17 ◁ C2×C16×C17 ◁ C2×C17×C32 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 2 generators on 145 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,28,25,32,49,31)(20,33,35,79,38,83,46,82)(21,39,42,95,45,99,107,98)(23,50,47,115,53,65,117,118)(24,54,48,30,56,71,93,77)(26,57,58,34,62,85,61,88)(27,63,66,109,69,128,68,43)(29,72,70,41,75,104,74,52)(36,90,89,135,91,131,113,141)(37,92,55,81,94,130,114,139)(40,73,100,138,86,78,105,137)(44,110,106,97,112,143,142,133)(51,80,120,132,84,64,116,129)(59,111,122,119,125,145,124,102)(60,87,123,121,126,140,76,101)(67,103,134,144,136,108,96,127);(18,20,25,38)(19,26,32,62)(21,40,45,86)(22,46,49,35)(23,51,53,84)(24,55,56,114)(27,64,69,80)(28,61,31,58)(29,73,75,78)(30,76,77,123)(33,39,83,99)(34,43,88,109)(36,41,91,52)(37,93,94,48)(42,105,107,100)(44,111,112,145)(47,116,117,120)(50,59,65,125)(54,60,71,126)(57,63,85,128)(66,132,68,129)(67,135,136,141)(70,137,74,138)(72,113,104,89)(79,98,82,95)(81,110,139,143)(87,127,140,144)(90,96,131,134)(92,97,130,133)(101,103,121,108)(102,142,119,106)(115,124,118,122)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 530);